Download 1 The Main Point Laws of Planetary Motion The Scientific Method

Lecture #3: Orbits and Gravity
The Main Point
• Laws of Planetary Motion:
Motions of planets, moons, and
asteroids can be very accurately
predicted because of the underlying
laws of planetary motion and gravity
discovered by Kepler and Newton.
– Kepler's Laws.
– Newton's Laws.
• Gravity.
• Planetary Orbits.
• Spacecraft Orbits.
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The Scientific Method
Laws of Planetary Motion
• Ptolemaic (geocentric) and Copernican
(heliocentric) systems made specific predictions
about where the planets would be over time.
• What was needed was high quality data to test
these competing models.
• Tycho Brahe (1546-1601):
– 20+ years of careful measurements.
– Best data available, and pre-telescopic!
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Laws of Planetary Motion
Kepler's First Law
• Each planet moves about the Sun in an orbit that is
an ellipse, with the Sun at one focus of the ellipse.
• Tycho provided the data, but his apprentice
provided the key new model.
• Johannes Kepler (1571-1630)
– Mathematician, theorist.
– Tycho's data weren't fit well by models
of Ptolemy or Copernicus.
– Worst case: Mars. WHY?
• Kepler's 20 years of studies led him to discover
three Laws of Planetary Motion.
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Fundamentals of an Ellipse
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Kepler's Second Law
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Kepler's Third Law
• The squares of the planets'
periods of revolution are in
direct proportion to the
cubes of the semimajor
axes of their orbits:
• The straight line joining a planet and the Sun
sweeps out equal areas in space in equal time.
p2 = a 3
where:
p = orbit period in years
a = semi-major axis, in AU
(1 AU = 150 million km)
("Astronomical Unit")
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Planetary Motions
Newton and Gravity
• Isaac Newton (1643-1727)
• Kepler's discoveries were spectacular!
• BUT... They are empirical laws only.
– Mathematician, philosopher.
– Synthesized observations and results
of Tycho, Kepler, and Galileo.
– Invented calculus!
– Formulated three Laws of Motion for all
objects (not just planets).
– Discovered the Universal Law of Gravity to explain the
motions of planets (and everything else!)
– Provided the framework for the later telescopic discovery of
Neptune, based on gravitational perturbations of Uranus' orbit.
• WHAT FORCES control this behavior?
• The answer was provided by Isaac Newton.
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Newton's Laws of Motion
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Newton's Laws of Motion
st
• Newton's 1 law really says momentum is conserved.
• First Law:
– Momentum = mass times velocity.
= mass times (speed, direction).
– For angular motion, momentum = mass times velocity
times distance from the axis of rotation.
– An object at rest remains at rest. An object in motion
remains moving uniformly in a straight line unless
acted on by an external force.
• Second Law:
– The change of motion of a body (acceleration) is
proportional to the force acting on it (F = ma).
• Third Law:
– To every action, there is an equal and opposite reaction.
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Newton's Laws of Motion
Other Fundamental Properties
• Conservation of Angular Momentum is a
fundamental concept in planetary science, to
which we will return many times...
• Mass:
– The amount of "stuff" in an object ("stuff" means protons
and neutrons and other subatomic particles) [grams].
– Mass ≠ Weight! Weight describes the force acting on mass.
• Volume:
– The physical space occupied by an object [cm3].
• Density:
– Mass / Volume = Mass per unit Volume [g/cm3]
– “How tightly packed”.
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Some typical densities
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Material
Neutron Star
Gold
Mercury
Lead
Iron
Earth (bulk)
Typical Rock
Water
Wood
Styrofoam peanut
Comet tail
Universe
Density (g/cm3)
~ 1015
19.3
13.6
11.4
7.9
Rocky planets
5.6
2.5
Gas giant
planets
1.0
0.8
0.1
~ 10-16
~ 10-29
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Gravity: The Tie that Binds
• Newton showed that gravity is the force that bends
the planets' straight line paths into ellipses.
• More generally, he showed that gravity is a
fundamental attractive force among all bodies
everywhere in space.
• Gravity obeys an inverse square law:
}
}
– Gravitational force = GM1M2 / R2.
– M1, M2 are two masses, R is their separation.
– G is a universal constant (Appendix A).
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Kepler Revisited
Planetary Orbits
• Newton found that Kepler's third law is actually a
simplified version of a more exact expression:
– If p in years, a in AU, and M in solar masses, then
p2 • (M1+M2) = a3
[notice: if M1=1 and M2 << M1 then p2 ≈ a3 ]
– Asteroid belt.
– Why there?
• A Powerful Result!
– If we can simply observe the period of rotation of one
object about another (a planet around the Sun, a satellite
around Jupiter, ...) and their separation, we can deduce
the mass of the system!
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• Generally circular.
• Most in same plane.
• Gap between Mars, Jupiter:
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Overhead view
• Many asteroids have high e.
• Most comets have high e, i.
• Is this arrangement typical?
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a
Object
(AU)
Mercury
0.39
Venus
0.72
Earth
1.00
Mars
1.52
1 Ceres
2.77
Jupiter
5.20
Saturn
9.54
76/P Halley 17.9
Uranus
19.2
Neptune
30.1
Pluto
39.5
1992 QB1 44.2
p
(YR)
0.24
0.62
1.00
1.88
4.60
11.9
29.5
76.0
84.1
164.8
248.6
359.4
C/Hyakutake 1003 32,000
e
(°)
0.21
0.01
0.02
0.09
0.08
0.05
0.06
0.97
0.05
0.01
0.25
0.08
0.99
i
(°)
7.0
3.4
0.0
1.85
10.6
1.31
2.49
162
0.78
1.78
17.2
2.2
125
(Appendix E)
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Oblique view
Edge-on view
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Summary
Kuiper Belt
• Kepler
– Discovered three empirical laws of planetary motion.
Oort Cloud
– Discovered three generalized laws of motion.
– Derived the basics of gravitational theory.
• Motions of planets, moons, asteroids, and comets
can be very accurately predicted!
• These discoveries laid the foundation for the
current Golden Age of spacecraft exploration.
Many more objects out there!
• 1 million+ asteroids?
• 1 trillion+ comets?
• The solar system does not
end at Neptune!
• More details: Lec. 32…
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• Newton
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Next Lecture...
• Seasons.
• Timekeeping.
• Calendars.
• Readings for next week:
– Chapters 2.2 and S1.
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